Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient and Practical Black-Box Verification of Quantum Metric Learning Algorithms

Ahmed Shokry, Movahhed Sadeghi, Mahmut Kandemir

Abstract

Quantum metric learning enhances machine learning by mapping classical data to a quantum Hilbert space with maximal separation between classes. However, on current NISQ hardware, this mapping process itself is prone to errors and could be fundamentally incorrect. Verifying that a quantum embedding model successfully achieves its promised separation is essential to ensure the correctness and reliability. In this paper, we propose a practical black-box verification protocol to audit the performance of quantum metric learning models. We define a setting with two parties: a powerful but untrusted prover, who claims to have a parameterized unitary circuit that embeds classical data from different groups with a guaranteed angular separation, and a limited verifier, whose quantum capabilities are restricted to performing only basic measurements. The verifier has no knowledge of the implementation of the prover, including the structure of the model, its parameters, or the details of the prover measurement setup. To verify the separation between different data groups, the proposed algorithm must overcome two key challenges. First, the verifier is ignorant of the prover's implementation details, such as the optimization cost function and measurement setup. Consequently, the verifier lacks any prior information about the expected quantum embedding states for each group. Second, the destructive nature of quantum measurements prevents direct estimation of the separation angles. Our algorithm successfully overcomes these challenges, enabling the verifier to accurately estimate the true separation angles between the different groups. We implemented the proposed protocol and deployed it to verify the QAOAEmbedding models. The results from both theoretical analysis and practical implementation show that our proposal effectively assesses embedding quality and remains robust in adversarial settings.

Efficient and Practical Black-Box Verification of Quantum Metric Learning Algorithms

Abstract

Quantum metric learning enhances machine learning by mapping classical data to a quantum Hilbert space with maximal separation between classes. However, on current NISQ hardware, this mapping process itself is prone to errors and could be fundamentally incorrect. Verifying that a quantum embedding model successfully achieves its promised separation is essential to ensure the correctness and reliability. In this paper, we propose a practical black-box verification protocol to audit the performance of quantum metric learning models. We define a setting with two parties: a powerful but untrusted prover, who claims to have a parameterized unitary circuit that embeds classical data from different groups with a guaranteed angular separation, and a limited verifier, whose quantum capabilities are restricted to performing only basic measurements. The verifier has no knowledge of the implementation of the prover, including the structure of the model, its parameters, or the details of the prover measurement setup. To verify the separation between different data groups, the proposed algorithm must overcome two key challenges. First, the verifier is ignorant of the prover's implementation details, such as the optimization cost function and measurement setup. Consequently, the verifier lacks any prior information about the expected quantum embedding states for each group. Second, the destructive nature of quantum measurements prevents direct estimation of the separation angles. Our algorithm successfully overcomes these challenges, enabling the verifier to accurately estimate the true separation angles between the different groups. We implemented the proposed protocol and deployed it to verify the QAOAEmbedding models. The results from both theoretical analysis and practical implementation show that our proposal effectively assesses embedding quality and remains robust in adversarial settings.

Paper Structure

This paper contains 17 sections, 4 theorems, 28 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Quantum Metric Learning
  4. Statistical Interactive Proofs
  5. Verification Procedure
  6. Formal Description and Challenges
  7. Protocol Details
  8. Completeness and Soundness Analysis
  9. Experiment Evaluation
  10. Experiment Setup
  11. Simulation Results
  12. Verifying QAOAEmbedding Model
  13. Discussion
  14. Extension to Multiple Groups
  15. Extension to Higher-Dimensional Feature Vectors
  16. ...and 2 more sections

Key Result

Lemma 1

Given two sets of quantum states $A = \{\ket{a_i}\}_{i=1}^{M_a}$ and $B = \{\ket{b_j}\}_{j=1}^{M_b}$ prepared by a parameterized quantum circuit $U(\vec{x}, \theta)\ket{0}$, under the optimization procedure that: The optimal solution achieves the following:

Figures (5)

  • Figure 1: Example of quantum metric learning.
  • Figure 2: The proposed verification protocol.
  • Figure 3: Angle estimation quality for the different theoretical models.
  • Figure 4: Effect of the number of samples $N$ on angle estimation.
  • Figure 5: Effect of the number of samples $N$ on fidelity.

Theorems & Definitions (8)

  • Lemma 1: Optimal Quantum Metric Learning
  • proof
  • Lemma 2: Sufficiency
  • proof
  • Lemma 3: Completeness
  • proof
  • Lemma 4: Soundness
  • proof